Proyecto FONDECYT 1090456

Ciencias Básicas

Marzo 2009 - Marzo 2013

Raimund Bürger [P]:

Resumen:

PROPOSAL ABSTRACT: Must be clear and informative. Describe the main issues you plan to address,
including goals, methodology and expected outcomes. A good summary facilitates an adequate description
and understanding of what you intend to achieve. If selected, this abstract may be published in CONICYT’s
web page. The maximum length for this section is 1 page. (Arial or Verdana, font size 10).
It is well known that first-order, nonlinear conservation laws and some related partial differential equations,
such as strongly degenerate parabolic equations, possess discontinuous solutions, even for smooth
initial data. These solutions need to be defined in a weak sense, and require an additional selection criterion,
or entropy condition, to select the so-called entropy solution among several weak solutions. The commonly
accepted Kruˇzkov-type entropy solution framework includes well-posedness (existence, uniqueness and stability)
of scalar equations with flux functions that depend smoothly on the data, time, and spatial coordinates;
several types of numerical schemes (including finite volumes and front tracking) are known to converge to
the entropy solution. This well-documented body of knowledge forms the standard entropy solution theory.
Motivated by models from several applications, it is proposed to investigate the well-posedness, to develop
numerical schemes, and to simulate applications of conservation laws that are bot covered by the
standard theory. The planned research is subdivided into three research topics (RTI, II and III).
The first research topic (RTI) deals with conservation laws and related equations whose flux function
depends discontinuously on the spatial variable. This problem occurs e.g. in models of traffic flow with
heterogeneous road surface conditions, clarifier-thickeners, and two-phase flow in heterogeneous media.
The key issue is the treatment of jumps of the solution at flux discontinuities, which requires different
entropy jump conditions for different applications. Here, the appropiate theory does not emerge as a limiting
case of the standard theory of equations whose fluxes are smooth functions of their arguments.
Recent results for a spatially one-dimensional scalar equation involving two different fluxes defined to
either side of a given spatial position, and which intersect once in a particular simple way, include wellposedness
and convergence of an Enguist-Osher type numerical scheme. It is planned to obtain similar
results for flux curves that intersect in a more general way. Furthermore, it is planned to investigate entropy
solutions and develop numerical schemes for one-dimensional hyperbolic systems of conservation laws with
discontinuous flux (starting from a model of multi-species vehicular traffic), and for scalar conservation laws
with discontinuous flux in two space dimensions.
In the second research topic (RTII), conservation laws and degenerate parabolic equations whose flux
does not depend on only the local value of the solution, but non-locally the function solution as a whole, will
be considered. Two variants of the problem, which arise from a model of sedimentation and an aggregation
model in mathematical biology, respectively, will be analyzed. In the sedimentation model, the non-locality
is introduced by convolution with a kernel of compact support, and the resulting equation is approximated
by a (standard, i.e., local) diffusive-dispersive PDE. The properties of the limiting model arising from letting
the compact support tend to zero are of interest, since non-classical shocks are possibly produced. In the
aggregation model, the non-locality arises from evaluating the flux at the total mass of the sought quantity.
The third research topic (RTIII) is focused on scalar conservation laws with a non-convex flux function
with “non-entropy” solutions that satisfy conditions other than standard entropy conditions. This problem
is associated with non-entropic solutions and possibly non-classical shocks. The prototype equation that
gives rise to this problem is an that appears in models of traffic flow, and where a particular behavioristic
principle, the “driver’s ride impulse”, compels solutions that differ from the standard theory. A related
problem arises in a model of sedimentation. We intend to conduct a well-posedness analysis for these
applications (utilizing the mathematical theory of non-classical shocks), and to develop numerical schemes,
starting from experiments with front tracking and the Artifical Compression Method.